Back seat is two words when it is a noun phrase with the adjective back modifying the noun seat. It is one word—either backseat or back-seat, with a hyphen—when it functions as an adjective. For example, it is an adjective in phrases such as backseat driver and backseat passenger. When you’re in doubt about which form to use, try replacing it with seat in the back. If that phrase would work in its place, the back seat is the logical choice.
When it comes to the adjective, the hyphenated form (back-seat) is about as common as backseat in 21st-century English, so either is correct. For newer compounds such this, though, the less adventurous hyphenated forms are safer in formal writing.
Since the incident, Russell says he did not use bad language, but his backseat passenger gleefully says he did. [Labradorian]
She bravely manages to break free by climbing over the back seat and escaping through a rear passenger door. [Herald Sun]
Mom’s turns at the wheel made things even worse; he was a dreadful back-seat driver. [Illinois Times]
[H]ow can it not mean that African lives must take a back seat to the profits of some of the richest corporations in the world? [Globe and Mail]